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1D RELATIVISTIC CONFINED COULOMB PROBLEM FOR DIRAC EQUATION (Z < 137)

Matchonov Quvonchbek Po'lat o'g'li

Published Sep 30, 2023• 0 citations
Publisher

Abstract

Consider first the case of quantum particle confined in a impenetrable relativistic box of size . Such system can be described by stationary one dimensional Dirac equation which is given by (natural units, )
DOI
10.5281/zenodo.8394854

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Abstract The problem of a particle in a box is probably the simplest problem in quantum mechanics which allows for significant insight into the nature of quantum systems and thus is a cornerstone in the teaching of quantum mechanics. In relativistic quantum mechanics this problem allows also to highlight the implications of special relativity for quantum physics, namely the effect that spin has on the quantised energy spectra. To illustrate this point, we solve the problem of a spin zero relativistic particle in a one- and three-dimensional box using the Klein–Gordon equation in the Feshbach–Villars formalism. We compare the solutions and the energy spectra obtained with the corresponding ones from the Dirac equation for a spin one-half relativistic particle. We note the similarities and differences, in particular the spin effects in the relativistic energy spectrum. As expected, the non-relativistic limit is the same for both kinds of particles, since, for a particle in a box, the spin contribution to the energy is a relativistic effect.
The Dirac equation is a relativistic wave equation and was the first equation to capture spin in relativistic quantum mechanics. Here, the Dirac equation will be derived and solved for a particle in a spherical box potential. Comparisons to the non-relativistic Schrödinger equation as well as to a relativistically corrected Schrödinger equation will be made. Applications of the Dirac equation, such as the Bogoliubov model, are examined and discussed and it is validated that the Dirac equation can provide knowledge about elementary particles.
The problem of a relativistic spin 1/2 particle confined to a one-dimensional box is solved in a way that resembles closely the solution of the well known quantum-mechanical textbook problem of a non-relativistic particle in a box. The energy levels and probability density are computed and compared with the non-relativistic case. Resumo. O problema de uma partícula de spin 1/2 confinada por uma caixa a uma dimensão é resolvido de uma maneira muito semelhante à da resolução do problema de uma partícula no-relativista numa caixa referido em muitos livros introdutórios de Mecânica Quntâica. Os níveis de energia e a densidade de probabilidade são calculados e comparados com os valores não-relativistas.